Method for measuring the complex dielectric constant of a substance

ABSTRACT

The method for measuring the complex dielectric constant of a substance is based on calculations made in the frequency domain. An open ended rectangular waveguide terminates, at one end, in a non-resonant rectangular cavity, which is then filled with a substance of interest. An electromagnetic wave of known frequency is transmitted through the waveguide and the reflection coefficient at the interface between the waveguide and the cavity is measured by a network analyzer or the like. Another measurement is then made with a slight variation in depth of the cavity or a variation in frequency, and from these two measurements of the reflection coefficient, the complex dielectric constant of the substance of interest can be calculated.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to measurements of materialproperties, and particularly to an optical frequency-based method formeasuring the complex dielectric constant of a substance.

2. Description of the Related Art

The measurement of the complex dielectric constant and complexpermeability is required not only for scientific but also for industrialapplications. For example, areas in which knowledge of the properties ofmaterials at microwave frequencies is required include microwaveheating, biological effects of microwaves, and nondestructive testing.

Although it is known to make measurements of the complex dielectricconstant and complex permeability in the time domain, the calculationsinvolved are complex and are dependent upon time and energy intensiveFourier transformations due to the measured transient response. Further,this technique typically only works within a severely limited band offrequencies, dependent upon the time response of the initial pulse andits repetition frequency. It would be desirable to be able to measurethe complex dielectric constant directly in the frequency domain,allowing for measurements to be made over a wide range of frequencies,with calculations being relatively fast and simple.

Thus, a method for measuring the complex dielectric constant of asubstance addressing the aforementioned problems is desired.

SUMMARY OF THE INVENTION

Embodiments of methods for measuring a complex dielectric constant of asubstance are based on calculations made in the frequency domain. Anopen ended rectangular waveguide terminates, at one end, in anon-resonant rectangular cavity, which is then filled with a substanceof interest. An electromagnetic wave of a known frequency is transmittedthrough the waveguide and a reflection coefficient at the interfacebetween the waveguide and the cavity is measured by a network analyzeror the like. Another measurement is then made with a slight variation indepth of the cavity or a variation in frequency, and from these twomeasurements of the reflection coefficient, the complex dielectricconstant of the substance of interest can be calculated. The method formeasuring the complex dielectric constant of a substance is performed asfollows:

(a) providing an open ended rectangular waveguide having across-sectional width a and a cross-sectional height b;

(b) providing a non-resonant rectangular cavity in communication withone of the ends of the open ended rectangular waveguide, thenon-resonant rectangular cavity having a cross-sectional width g, across-sectional height h and a cross-sectional length d, thenon-resonant rectangular cavity being symmetrically fed anelectromagnetic wave by the open ended rectangular waveguide;

(c) filling the non-resonant rectangular cavity with a substance ofinterest, and it should be understood that the substance can be inpowdered, liquid or solid form, and the cavity is filled without bubblesor gaps in the filled cavity;

(d) transmitting an electromagnetic wave through the open endedrectangular waveguide toward the non-resonant rectangular cavity, anddesirably, only the dominant mode of the electromagnetic wave is allowedto propagate, and the source of the wave should be about 10λ from theopen end of the waveguide, where λ is the wavelength of theelectromagnetic wave;

(e) measuring a first reflection coefficient R₁ at an interface betweenthe open ended rectangular waveguide and the non-resonant rectangularcavity, and the magnitude and phase of the reflection coefficient can bemeasured by a conventional network analyzer or the like, and themeasurement location is desirably a distance of at least 10λ, from theinterface;

(f) varying the cross-sectional length d of the non-resonant rectangularcavity by a length δ and, as an alternative to varying thecross-sectional length, the frequency of the electromagnetic wave can bevaried;

(g) re-transmitting the electromagnetic wave through the open endedrectangular waveguide toward the non-resonant rectangular cavity;

(h) measuring a second reflection coefficient R₂ at the interfacebetween the open ended rectangular waveguide and the non-resonantrectangular cavity;

(i) calculating a first variable LL as

${{LL} = {{\sum\limits_{m = 1}\;{\sum\limits_{n = 1}\;{{B_{mn}\left( \frac{n\;\pi}{b} \right)}{I_{yg}\left( {m,n} \right)}}}} - {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{m = 1}\;{\sum\limits_{n = 1}\;{A_{mn}{\gamma_{mn}\left( \frac{m\;\pi}{a} \right)}{I_{yg}\left( {m,n} \right)}}}}} - {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{m = 2}\;{A_{m\; 0}{\gamma_{mo}\left( \frac{m\;\pi}{a} \right)}{I_{yg}\left( {m,o} \right)}}}}}},$where m and n are summation indices ranging between 1 and N, where N isan integer selected for stabilization of the summations,

${A_{mn} = {\frac{{- m}\;{\pi/a}}{k_{mn}^{2}}\left( \frac{4}{ab} \right){I_{yg}\left( {m,n} \right)}}},$where

$k_{mn} = \sqrt{\left( \frac{m\;\pi}{a} \right)^{2} + \left( \frac{n\;\pi}{b} \right)^{2}}$and

${{I_{yg}\left( {m,n} \right)} = {\int_{x = 0}^{a}{\int_{y = 0}^{b}{{f_{y}\left( {x,y} \right)}{\sin\left( \frac{m\;\pi\; x}{a} \right)}{\cos\left( \frac{n\;\pi\; y}{b} \right)}\ {\mathbb{d}x}\ {\mathbb{d}y}}}}},$where

${{f_{y}\left( {x,y} \right)} = \frac{\sin\left( {\frac{\pi}{a}x} \right)}{\sqrt{1 - \left( \frac{y - \frac{b}{2}}{\frac{b}{2}} \right)^{2}}}},$x and y being Cartesian coordinates corresponding to width and height,respectively,

${B_{mn} = {{- \left( \frac{\frac{n\;\pi}{b}}{\frac{m\;\pi}{a}} \right)}\left( \frac{j\;{\omega\varepsilon}_{o}}{\gamma_{mn}} \right)A_{mn}}},$where j is the imaginary number, ω is an angular frequency of theelectromagnetic wave, ∈_(o) is the constant permittivity of free space,and

${\gamma_{mn} = \sqrt{k_{mn}^{2} - k_{o}^{2}}},$where

${k_{0} = {\omega\sqrt{\mu_{0}\varepsilon_{0}}}},$where μ₀ the constant magnetic permeability of free space, wherein

${A_{m\; 0} = {{\frac{- 2}{m\;{\pi/a}}\left( \frac{1}{ab} \right){I_{yg}\left( {m,0} \right)}\mspace{14mu}{for}\mspace{14mu} m} \neq 1}};$

(j) establishing a first set of estimated values for a real part of adielectric constant associated with the substance of interest, ∈_(i),where i ranges between 1 and a pre-selected integer S;

(k) calculating a second variable M_(i) for i ranging between 1 and S as

${M_{i} = {{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{{C_{pq}\left( \frac{q\;\pi}{h} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{D_{pq}{\alpha_{pq}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{D_{p\; 0}{\alpha_{po}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{p\; 0}d} \right)}{I_{yc}\left( {p,o} \right)}}}}}},$where p and q are summation indices ranging between 1 and N,

${D_{pq} = {\frac{{- p}\;{\pi/g}}{k_{pq}^{2}{\sin\left( {\alpha_{pq}d} \right)}}\left( \frac{4}{gh} \right){I_{yc}\left( {p,q} \right)}}},$where

$k_{pq} = \sqrt{\left( \frac{p\;\pi}{g} \right)^{2} + \left( \frac{q\;\pi}{h} \right)^{2}}$and

${\alpha_{pq} = \sqrt{k_{i}^{2} - k_{pq}^{2}}},$where

${k_{i} = {\omega\sqrt{\mu_{0}\varepsilon_{o}ɛ_{i}}}},$and

${C_{pq} = {{- \left( \frac{\frac{q\;\pi}{h}}{\frac{p\;\pi}{g}} \right)}\left( \frac{\left( {{j\;{\omega ɛ}_{o}ɛ_{i}} + \delta} \right)}{\alpha_{pq}} \right)D_{pq}}},$where

${{I_{yc}\left( {p,q} \right)} = {\int_{x = 0}^{a}{\int_{y = 0}^{b}{{f_{y}\left( {x,y} \right)}{\sin\left( \frac{p\;\pi\; x}{g} \right)}{\cos\left( \frac{q\;\pi\; y}{h} \right)}\ {\mathbb{d}x}\ {\mathbb{d}y}\mspace{14mu}{and}}}}}\mspace{14mu}$${D_{p\; 0} = {\frac{- 1}{\frac{p\;\pi}{g}{\sin\left( {\alpha_{p\; 0}d} \right)}}\left( \frac{2}{gh} \right){I_{yc}\left( {p,0} \right)}}};$

(l) establishing a first set of values of ∈_(i) corresponding to thecalculated second variable M_(i) which satisfy the condition

${\frac{1 - R_{1}}{1 + R_{1}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}};$

(m) establishing a second set of values of ∈_(i) corresponding to thecalculated second variable M_(i) which satisfy the condition

${\frac{1 - R_{2}}{1 + R_{2}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}};$

(n) selecting a value of ∈_(i) which is a member of both the first setand the second set of values corresponding to the calculated secondvariable M_(i) and setting this selected value equal to a variable∈_(d);

(o) establishing a second set of estimated values for the real part of adielectric constant associated with the substance of interest, ∈_(ri),where i ranges between 1 and S, such that ∈_(ri)=∈_(i)−j∈′, where ∈′ isan imaginary part of the dielectric constant associated with thesubstance of interest, where ∈_(r1)=∈_(d)−x₁ and ∈_(rS)=∈_(d)+x₁ and∈′=x₂, where x₁ is approximately equal to 0.5 and x₂ is a pre-selectedvalue based on expected conductivity of the substance of interest;

(p) re-calculating the second variable M_(i) for i ranging between 1 andS as M_(i)=

${{{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{{C_{pq}\left( \frac{q\;\pi}{h} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{D_{pq}{\alpha_{pq}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{D_{p\; 0}{\alpha_{po}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{p\; 0}d} \right)}{I_{yc}\left( {p,o} \right)}}}}},\mspace{79mu}{{{where}\mspace{14mu} k_{i}} = {\omega\sqrt{\mu_{0}{\varepsilon_{0}\left( {ɛ_{ri} - ɛ^{\prime}} \right)}}}}}\mspace{14mu}$$\mspace{79mu}{{{{and}\mspace{14mu} C_{pq}} = {{- \left( \frac{\frac{q\;\pi}{h}}{\frac{p\;\pi}{g}} \right)}\left( \frac{\left( {{j\;{\omega ɛ}_{o}ɛ_{ri}} + \delta} \right)}{\alpha_{pq}} \right)D_{pq}}};}$

(q) establishing a first set of values of ∈_(ri) corresponding to there-calculated second variable M_(i) which satisfy the condition

${\frac{1 - R_{1}}{1 + R_{1}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}};$

(r) establishing a second set of values of ∈_(ri) corresponding to there-calculated second variable M_(i) which satisfy the condition

${\frac{1 - R_{2}}{1 + R_{2}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}};$

(s) selecting a value of ∈_(ri) which is a member of both the first setand the second set of values corresponding to the re-calculated secondvariable M_(i) and setting the selected value equal to a variable∈_(rd); and

(t) re-calculating the imaginary part of the dielectric constantassociated with the substance of interest, ∈′, as ∈′=j(∈_(rd)−∈_(d)),wherein the dielectric constant associated with the substance ofinterest is determined as ∈_(rd)+j(∈_(rd)−∈_(d)).

These and other features of the present invention will become readilyapparent upon further review of the following specification.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 diagrammatically illustrates an optical configuration forperforming the method for measuring the complex dielectric constant of asubstance according to the present invention.

FIG. 2 is a block diagram illustrating system components of a system forperforming calculations associated with the method for measuring thecomplex dielectric constant of a substance, and an associated networkanalyzer and electromagnetic wave generator, according to the presentinvention.

Unless otherwise indicated, similar reference characters denotecorresponding features consistently throughout the attached drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Embodiments of methods for measuring the complex dielectric constant ofa substance are based on calculations made in the frequency domain. Asshown in the overall configuration 10 of FIG. 1, an open endedrectangular waveguide 12 terminates, at one end, in a non-resonantrectangular cavity 14, which is then filled with a substance ofinterest. An electromagnetic wave W of known frequency, such asgenerated by an electromagnetic wave generator 122, as can be a suitableconventional electromagnetic wave generator, is transmitted through thewaveguide 12 and the reflection coefficient at the interface I betweenthe waveguide 12 and the cavity 14 is measured by a network analyzer120, such as a conventional network analyzer or the like, for example.Another measurement is then made with a slight variation in depth of thecavity or a variation in frequency and, from these two measurements ofthe reflection coefficient, the complex dielectric constant of thesubstance of interest can be calculated.

An embodiment of a method for measuring the complex dielectric constantof a substance includes performing the following steps of operations.The open ended rectangular waveguide 12 is provided for the dielectricconstant measurement, the open ended rectangular waveguide 12 having across-sectional width a and a cross-sectional height b. Also, thenon-resonant rectangular cavity 14 is provided in communication with oneof the ends of the open ended rectangular waveguide 12, the non-resonantrectangular cavity 14 having a cross-sectional width g, across-sectional height h and a cross-sectional length d, thenon-resonant rectangular cavity 14 being symmetrically fed anelectromagnetic wave by the open ended rectangular waveguide 12.

Then, the non-resonant rectangular cavity 14 is filled with a substanceof interest. It should be understood that the substance of interest canbe in powdered, liquid or solid form, but the non-resonant rectangularcavity 14 is filled without bubbles or gaps in the filled non-resonantrectangular cavity 14. An electromagnetic wave W is transmitted throughthe open ended rectangular waveguide 12 toward the non-resonantrectangular cavity 14. Desirably, only the dominant mode of theelectromagnetic wave W is allowed to propagate. The source of the wave Wshould be about 10λ from the open end of the waveguide, where λ is thewavelength of the electromagnetic wave W, such as can be in meters, andthe wave W can be generated by a suitable wave generator, such as theelectromagnetic wave generator 122, for example.

Then a first reflection coefficient R₁ is measured at the interface Ibetween the open ended rectangular waveguide 12 and the non-resonantrectangular cavity 14. The magnitude and phase of the reflectioncoefficient can be measured by the network analyzer 120, such as aconventional network analyzer, such as a Vector Network Analyzer (VNA)or a network analyzer using a standard Null Shift and a VSWA technique,or the like, for example, and the measurement location is desirably adistance of at least 10λ from the interface I. The cross-sectionallength d of the non-resonant rectangular cavity 14 is varied by a lengthδ. As an alternative to varying the cross-sectional length, thefrequency of electromagnetic wave W can be varied for the measurement,for example. The electromagnetic wave W is re-transmitted through theopen ended rectangular waveguide 12 toward the non-resonant rectangularcavity 14 and a second reflection coefficient R₂ is measured by thenetwork analyzer 120 at the interface I between the open endedrectangular waveguide 12 and the non-resonant rectangular cavity 14.

Then there is calculated, such as by system 100, a first variable LL as

${{LL} = {{\sum\limits_{m = 1}\;{\sum\limits_{n = 1}\;{{B_{mn}\left( \frac{n\;\pi}{b} \right)}{I_{yg}\left( {m,n} \right)}}}} - {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{m = 1}\;{\sum\limits_{n = 1}\;{A_{mn}{\gamma_{mn}\left( \frac{m\;\pi}{a} \right)}{I_{yg}\left( {m,n} \right)}}}}} - {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{m = 2}\;{A_{m\; 0}{\gamma_{mo}\left( \frac{m\;\pi}{a} \right)}{I_{yg}\left( {m,o} \right)}}}}}},$where m and n are summation indices ranging between 1 and N, where N isan integer selected for stabilization of the summations,

${A_{mn} = {\frac{{- m}\;{\pi/a}}{k_{mn}^{2}}\left( \frac{4}{ab} \right){I_{yg}\left( {m,n} \right)}}},$where

$k_{mn} = \sqrt{\left( \frac{m\;\pi}{a} \right)^{2} + \left( \frac{n\;\pi}{b} \right)^{2}}$and

${{I_{yg}\left( {m,n} \right)} = {\int_{x = 0}^{a}{\int_{y = 0}^{b}{{f_{y}\left( {x,y} \right)}{\sin\left( \frac{m\;\pi\; x}{a} \right)}{\cos\left( \frac{n\;\pi\; y}{b} \right)}\ {\mathbb{d}x}\ {\mathbb{d}y}}}}},{{{where}\mspace{14mu}{f_{y}\left( {x,y} \right)}} = \frac{\sin\left( {\frac{\pi}{a}x} \right)}{\sqrt{1 - \left( \frac{y - \frac{b}{2}}{\frac{b}{2}} \right)^{2}}}},$x and y being Cartesian coordinates corresponding to width and height,respectively,

${B_{mn} = {{- \left( \frac{\frac{n\;\pi}{b}}{\frac{m\;\pi}{a}} \right)}\left( \frac{j\;{\omega\varepsilon}_{o}}{\gamma_{mn}} \right)A_{mn}}},$where j is the imaginary number, co is an angular frequency of theelectromagnetic wave, ∈_(o) is the constant permittivity of free space,and

${\gamma_{mn} = \sqrt{k_{mn}^{2} - k_{o}^{2}}},$where

${k_{0} = {\omega\sqrt{\mu_{0}\varepsilon_{o}}}},$where μ₀ is the constant magnetic permeability of free space, typicallyequal to 4π×10⁷, wherein

$A_{m\; 0} = {{\frac{- 2}{m\;{\pi/a}}\left( \frac{1}{ab} \right){I_{yg}\left( {m,0} \right)}\mspace{14mu}{for}\mspace{14mu} m} \neq 1.}$It should be understood that N is suitably large to ensure stabilizationof the summations, for example.

Then, a first set of estimated values is established for a real part ofa dielectric constant associated with the substance of interest, ∈_(i),where i ranges between 1 and a pre-selected integer S. A second variableM_(i) is then calculated, such as by the system 100, for i rangingbetween 1 and S as

${M_{i} = {{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{{C_{pq}\left( \frac{q\;\pi}{h} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{D_{pq}{\alpha_{pq}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{D_{p\; 0}{\alpha_{po}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{p\; 0}d} \right)}{I_{yc}\left( {p,o} \right)}}}}}},$where p and q are summation indices ranging between 1 and N,

${D_{pq} = {\frac{{- p}\;{\pi/g}}{k_{pq}^{2}{\sin\left( {\alpha_{p\; q}d} \right)}}\left( \frac{4}{gh} \right){I_{yc}\left( {p,q} \right)}}},$where

$k_{pq} = \sqrt{\left( \frac{p\;\pi}{g} \right)^{2} + \left( \frac{q\;\pi}{h} \right)^{2}}$and

${\alpha_{pq} = \sqrt{k_{i}^{2} - k_{pq}^{2}}},$where

${k_{i} = {\omega\sqrt{\mu_{0}\varepsilon_{o}ɛ_{i}}}},$and

${C_{pq} = {{- \left( \frac{\frac{q\;\pi}{h}}{\frac{p\;\pi}{g}} \right)}\left( \frac{\left( {{j\;{\omega ɛ}_{o}ɛ_{i}} + \delta} \right)}{\alpha_{pq}} \right)D_{pq}}},$where

${{I_{yc}\left( {p,q} \right)} = {\int_{x = 0}^{a}{\int_{y = 0}^{b}{{f_{y}\left( {x,y} \right)}{\sin\left( \frac{p\;\pi\; x}{g} \right)}{\cos\left( \frac{q\;\pi\; y}{h} \right)}\ {\mathbb{d}x}\ {\mathbb{d}y}\mspace{14mu}{and}}}}}\mspace{14mu}$$D_{p\; 0} = {\frac{- 1}{\frac{p\;\pi}{g}{\sin\left( {\alpha_{p\; 0}d} \right)}}\left( \frac{2}{gh} \right){{I_{yc}\left( {p,0} \right)}.}}$As noted above, N is suitably large to ensure stabilization of thesummations.

Based upon the above calculations, the measurement of the firstreflection coefficient R₁ and the measurement of the second reflectioncoefficient R₂, a first set of values of ∈_(i) corresponding to thecalculated second variable M_(i) is established which satisfy thecondition

$\frac{1 - R_{1}}{1 + R_{1}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}$and a second set of values of ∈_(i) corresponding to the calculatedsecond variable M_(i) is established which satisfy the condition

$\frac{1 - R_{2}}{1 + R_{2}} = {\frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}.}$It should be understood that more than one value of ∈_(i) can be foundwhich satisfies this equation, thus the first and second sets can havemultiple members, for example.

Then a value of ∈_(i) is selected which is a member of both the firstset and the second set of values corresponding to the calculated secondvariable M_(i) and this selected value is set equal to a variable ∈_(d).Based on the selected value equal to the variable ∈_(d), a second set ofestimated values is established for the real part of a dielectricconstant associated with the substance of interest, ∈_(ri), where iranges between 1 and S, such that ∈_(ri)=∈_(i)−j∈′, where ∈′ is animaginary part of the dielectric constant associated with the substanceof interest, where ∈_(r1)=∈_(d)−x₁ and ∈_(rS)=∈_(d)+x₁ and ∈′=x₂, wherex₁ is approximately equal to 0.5 and x₂ is a pre-selected value based onexpected conductivity of the substance of interest. The variable x₂ranges between zero and a suitable value dependent on the expectedconductivity of the substance, for example. Any value can be assumed forx₂, and if no solution can be found, this value for x₂ can be increased.

Then, using the second set of estimated values that are based on theselected value equal to the variable ∈_(d), the second variable M_(i) isre-calculated for i ranging between 1 and S as

${{M_{i} = {{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{{C_{pq}\left( \frac{q\;\pi}{h} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{D_{pq}{\alpha_{pq}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{D_{p\; 0}{\alpha_{po}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{p\; 0}d} \right)}{I_{yc}\left( {p,o} \right)}}}}}},\mspace{79mu}{{{where}\mspace{14mu} k_{i}} = {\omega\sqrt{\mu_{0}{\varepsilon_{o}\left( {ɛ_{ri} - ɛ^{\prime}} \right)}}\mspace{14mu}{and}}}}\mspace{14mu}$$\mspace{79mu}{C_{pq} = {{- \left( \frac{\frac{q\;\pi}{h}}{\frac{p\;\pi}{g}} \right)}\left( \frac{\left( {{j\;{\omega ɛ}_{o}ɛ_{ri}} + \delta} \right)}{\alpha_{pq}} \right){D_{pq}.}}}$

Using the re-calculated second variable M_(i), a first set of values of∈_(ri) corresponding to the re-calculated second variable M_(i) isestablished which satisfy the condition

$\frac{1 - R_{1}}{1 + R_{1}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}$and a second set of values of ∈_(ri) corresponding to the re-calculatedsecond variable M_(i) is established which satisfy the condition

$\frac{1 - R_{2}}{1 + R_{2}} = {\frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}.}$Then a value of ∈_(ri) is selected, such as by the system 100, which isa member of both the first set and the second set of valuescorresponding to the re-calculated second variable M_(i), and theselected value is set equal to a variable ∈_(rd). The imaginary part ofthe dielectric constant associated with the substance of interest, ∈′,is re-calculated, such as by the system 100, as ∈′=j(∈_(rd)−∈_(d)), andthe dielectric constant associated with the substance of interest isdetermined, such as by the system 100, as ∈_(rd)+j(∈_(rd)−∈_(d)).

An apparatus to measure the complex dielectric constant of a substanceof interest is also provided, such as illustrated in FIGS. 1 and 2, forexample. The apparatus, generally illustrated by the configuration 10 ofFIG. 1 and the system 100 of FIG. 2, includes the open ended rectangularwaveguide 12, the non-resonant rectangular cavity 14 and the system 100.The system 100 of the apparatus includes a processor 114 and a memory112, the processor 114 executing instructions of a program stored in thememory 112 to implement the determination of the complex dielectricconstant, as described above, transforming the above described data andmeasurement information by implementing the above described algorithms,instructions, operations, and process steps to provide the determinedcomplex dielectric constant of the substance of interest.

The apparatus can also have an associated electromagnetic wavegenerator, such as the electromagnetic wave generator 122, to generateand transmit the electromagnetic wave W through the open endedrectangular waveguide 12 toward the non-resonant rectangular cavity 14.Also, the apparatus can have associated suitable detectors, such as thenetwork analyzer 120, or the like, to detect and measure the firstreflection coefficient R₁ of the transmitted electromagnetic wave Wmeasured at the interface I between the open ended rectangular waveguide12 and the non-resonant rectangular cavity 14 and to detect and measurethe second reflection coefficient R₂ measured at the interface I betweenthe open ended rectangular waveguide and the non-resonant rectangularcavity 14 and to provide data and measurement information thereon to thesystem 100.

Further, the non-resonant rectangular cavity 14 can have an adjustableand variable cross-sectional length d so as to be adjustable to have afirst cross-sectional length d and a second cross-sectional length ddifferent from the first cross-sectional length d. Such adjustment ofthe cross-sectional length d can be provided by suitable adjustmentmembers and/or associated with adjustable sides in the non-resonantrectangular cavity 14, to vary the dimensions or dimensional size of thenon-resonant rectangular cavity 14, for example,

In the above, the measurement equipment or apparatus typically requiredis a computer-controlled network analyzer for measuring the reflectioncoefficient. It should be understood that any suitable type of networkanalyzer or similar system can be used, as described. It should befurther understood that the calculations for the determination of thedielectric constant can be performed by any suitable computer system,such as the system 100 that is diagrammatically shown in FIG. 2. Datafor determining of the complex dielectric constant and programs orinstructions for implementing the measurements and calculations fordetermination of the complex dielectric constant are entered into thesystem 100 via any suitable type of interface 116, and can be stored inthe memory 112, which can be any suitable type of computer readable andprogrammable memory and is desirably a non-transitory, computer readablestorage medium. Calculations are performed by the processor 114, whichcan be any suitable type of computer processor and can be displayed tothe user on a display 118, which can be any suitable type of computerdisplay.

The processor 114 can be associated with, or incorporated into, anysuitable type of computing device, for example, a personal computer or aprogrammable logic controller (PLC) or an application specificintegrated circuit (ASIC). The display 118, the processor 114, thememory 112 and any associated computer readable recording media are incommunication with one another by any suitable type of data bus, as iswell known in the art.

Examples of computer-readable recording media include non-transitorystorage media, a magnetic recording apparatus, an optical disk, amagneto-optical disk, and/or a semiconductor memory (for example, RAM,ROM, etc.). Examples of magnetic recording apparatus that can be used inaddition to the memory 112, or in place of the memory 112, include ahard disk device (HDD), a flexible disk (FD), and a magnetic tape (MT).Examples of the optical disk include a DVD (Digital Versatile Disc), aDVD-RAM, a CD-ROM (Compact Disc-Read Only Memory), and a CD-R(Recordable)/RW. It should be understood that non-transitorycomputer-readable storage media include all computer-readable media,with the sole exception being a transitory, propagating signal.

It is to be understood that the present invention is not limited to theembodiments described above, but encompasses any and all embodimentswithin the scope of the following claims.

We claim:
 1. A method for measuring a complex dielectric constant of asubstance, comprising the steps of: (a) providing an open endedrectangular waveguide having a cross-sectional width a and across-sectional height b; (b) providing a non-resonant rectangularcavity in communication with one of the ends of the open endedrectangular waveguide, said non-resonant rectangular cavity having across-sectional width g, a cross-sectional height h and across-sectional length d, said non-resonant rectangular cavity beingsymmetrically fed by an electromagnetic wave by the open endedrectangular waveguide; (c) filling the non-resonant rectangular cavitywith a substance of interest; (d) transmitting an electromagnetic wavegenerated by an electromagnetic wave generator through the open endedrectangular waveguide toward the non-resonant rectangular cavity; (e)measuring a first reflection coefficient R₁ at an interface between theopen ended rectangular waveguide and the non-resonant rectangular cavitywith a network analyzer; (f) varying the cross-sectional length d of thenon-resonant rectangular cavity by a length ∈; (g) re-transmitting theelectromagnetic wave through the open ended rectangular waveguide towardthe non-resonant rectangular cavity; (h) measuring a second reflectioncoefficient R₂ at the interface between the open ended rectangularwaveguide and the non-resonant rectangular cavity with the networkanalyzer; (i) calculating a first variable LL as${{LL} = {{\sum\limits_{m = 1}\;{\sum\limits_{n = 1}\;{{B_{mn}\left( \frac{n\;\pi}{b} \right)}{I_{yg}\left( {m,n} \right)}}}} - {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{m = 1}\;{\sum\limits_{n = 1}\;{A_{mn}{\gamma_{mn}\left( \frac{m\;\pi}{a} \right)}{I_{yg}\left( {m,n} \right)}}}}} - {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{m = 2}\;{A_{m\; 0}{\gamma_{mo}\left( \frac{m\;\pi}{a} \right)}{I_{yg}\left( {m,o} \right)}}}}}},$where m and n are summation indices ranging between 1 and N, where N isan integer selected for stabilization of the summations,${A_{mn} = {\frac{{- m}\;{\pi/a}}{k_{mn}^{2}}\left( \frac{4}{ab} \right){I_{yg}\left( {m,n} \right)}}},$where$k_{mn} = \sqrt{\left( \frac{m\;\pi}{a} \right)^{2} + \left( \frac{n\;\pi}{b} \right)^{2}}$and${{I_{yg}\left( {m,n} \right)} = {\int_{x = 0}^{a}{\int_{y = 0}^{b}{{f_{y}\left( {x,y} \right)}{\sin\left( \frac{m\;\pi\; x}{a} \right)}{\cos\left( \frac{n\;\pi\; y}{b} \right)}\ {\mathbb{d}x}\ {\mathbb{d}y}}}}},$where${{f_{y}\left( {x,y} \right)} = \frac{\sin\left( {\frac{\pi}{a}x} \right)}{\sqrt{1 - \left( \frac{y - \frac{b}{2}}{\frac{b}{2}} \right)^{2}}}},$x and y being Cartesian coordinates corresponding to width and height,respectively,${B_{mn} = {{- \left( \frac{\frac{n\;\pi}{b}}{\frac{m\;\pi}{a}} \right)}\left( \frac{j\;{\omega\varepsilon}_{o}}{\gamma_{mn}} \right)A_{mn}}},$where j is the imaginary number, ω is an angular frequency of theelectromagnetic wave, ∈_(o) is the constant permittivity of free space,and ${\gamma_{mn} = \sqrt{k_{mn}^{2} - k_{o}^{2}}},$ where${k_{0} = {\omega\sqrt{\mu_{0}\varepsilon_{o}}}},$ where μ₀ is theconstant magnetic permeability of free space, wherein$A_{m\; 0} = {\frac{- 2}{m\;{\pi/a}}\left( \frac{1}{ab} \right){I_{yg}\left( {m,0} \right)}}$for m≠1; (j) establishing a first set of estimated values for a realpart of a dielectric constant associated with the substance of interest,∈_(i), where i ranges between 1 and a pre-selected integer S; (k)calculating a second variable M_(i) for i ranging between 1 and S as${M_{i} = {{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{{C_{pq}\left( \frac{q\;\pi}{h} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{\sum\limits_{q = 1}\;{D_{pq}{\alpha_{pq}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}\;{D_{p\; 0}{\alpha_{po}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{p\; 0}d} \right)}{I_{yc}\left( {p,o} \right)}}}}}},$where p and q are summation indices ranging between 1 and N,${D_{pq} = {\frac{{- p}\;{\pi/g}}{k_{pq}^{2}{\sin\left( {\alpha_{pq}d} \right)}}\left( \frac{4}{gh} \right){I_{yc}\left( {p,q} \right)}}},$where$k_{pq} = \sqrt{\left( \frac{p\;\pi}{g} \right)^{2} + \left( \frac{q\;\pi}{h} \right)^{2}}$and ${\alpha_{pq} = \sqrt{k_{i}^{2} - k_{pq}^{2}}},{where}$${k_{i} = {\omega\sqrt{\mu_{0}\varepsilon_{o}ɛ_{i}}}},$ and${C_{pq} = {{- \left( \frac{\frac{q\;\pi}{h}}{\frac{p\;\pi}{g}} \right)}\left( \frac{\left( {{{j\omega}_{o}ɛ_{i}} + \delta} \right)}{\alpha_{pq}} \right)D_{pq}}},$where${I_{yc}\left( {p,q} \right)} = {\int_{x = 0}^{a}{\int_{y = 0}^{b}{{f_{y}\left( {x,y} \right)}{\sin\left( \frac{p\;\pi\; x}{g} \right)}{\cos\left( {\frac{q\;\pi\; y}{h}\ {\mathbb{d}x}\ {\mathbb{d}y}} \right)}}}}$and${D_{p\; 0} = {\frac{- 1}{\frac{p\;\pi}{g}{\sin\left( {\alpha_{p\; 0}d} \right)}}\left( \frac{2}{g\; h} \right){I_{yc}\left( {p,0} \right)}}};$(l) establishing a first set of values of ∈_(i) corresponding to thecalculated second variable M_(i) which satisfy the condition${\frac{1 - R_{1}}{1 + R_{1}} = \frac{\left( {{LL} - M_{i}} \right)({ab}){j\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}};$(m) establishing a second set of values of ∈_(i) corresponding to thecalculated second variable M_(i) which satisfy the condition${\frac{1 - R_{2}}{1 + R_{2}} = \frac{\left( {{LL} - M_{i}} \right)({ab}){j\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}};$(n) selecting a value of ∈_(i) which is a member of both the first setand the second set of values corresponding to the calculated secondvariable M_(i) and setting this selected value equal to a variable∈_(d); (o) establishing a second set of estimated values for the realpart of the dielectric constant associated with the substance ofinterest, ∈_(ri), where i ranges between 1 and S, such that∈_(ri)=∈_(i)−j∈′, where ∈′ is an imaginary part of the dielectricconstant associated with the substance of interest, where∈_(r1)=∈_(d)−X₁ and ∈_(rS)=∈_(d)+x₁ and ∈′=x₂, where x₁ is approximatelyequal to 0.5 and x₂ is a pre-selected value based on expectedconductivity of the substance of interest; (p) re-calculating the secondvariable M_(i) for i ranging between 1 and S as${M_{i} = {{\sum\limits_{p = 1}{\sum\limits_{q = 1}^{\;}{{C_{pq}\left( \frac{q\;\pi}{h} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}} + {\frac{1}{{j\omega\mu}_{o}}{\sum\limits_{p = 1}^{\;}{\sum\limits_{q = 1}^{\;}{D_{pg}{\alpha_{pq}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}}} + {\frac{1}{{j\omega\mu}_{o}}{\sum\limits_{p = 1}^{\;}{D_{p\; 0}{\alpha_{po}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{p\; 0}d} \right)}{I_{yc}\left( {p,o} \right)}}}}}},$where$k_{i} = {\omega\sqrt{\mu_{0}{\varepsilon_{o}\left( {ɛ_{ri} - ɛ^{\prime}} \right)}}}$and${C_{pq} = {{- \left( \frac{\frac{q\;\pi}{h}}{\frac{p\;\pi}{g}} \right)}\left( \frac{\left( {{{j\omega ɛ}_{o}ɛ_{ri}} + \delta} \right)}{\alpha_{p\; q}} \right)D_{pq}}};$(q) establishing a first set of values of ∈_(ri) corresponding to there-calculated second variable M_(i) which satisfy the condition${\frac{1 - R_{1}}{1 + R_{1}} = \frac{\left( {{LL} - M_{i}} \right)({ab}){j\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}};$(r) establishing a second set of values of ∈_(ri) corresponding to there-calculated second variable M_(i) which satisfy the condition${\frac{1 - R_{2}}{1 + R_{2}} = \frac{\left( {{LL} - M_{i}} \right)({ab}){j\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}};$(s) selecting a value of ∈_(ri) which is a member of both the first setand the second set of values corresponding to the re-calculated secondvariable M_(i) and setting the selected value equal to a variable∈_(rd); and (t) re-calculating the imaginary part of the dielectricconstant associated with the substance of interest, ∈′, as∈′=j(∈_(rd)−∈_(d)), wherein the dielectric constant associated with thesubstance of interest is determined as ∈_(rd)+j(∈_(rd)−∈_(d)).
 2. Anapparatus to measure a complex dielectric constant of a substance,comprising: an open ended rectangular waveguide having a cross-sectionalwidth a and a cross-sectional height b adapted to receive anelectromagnetic wave through an open end thereof; an electromagneticwave generator for generating the electromagnetic wave; a non-resonantrectangular cavity in communication with one of the ends of the openended rectangular waveguide at an interface between the open endedrectangular waveguide and the non-resonant rectangular cavity, saidnon-resonant rectangular cavity having a cross-sectional width g, across-sectional height h and selectively adjustable a cross-sectionallength d, said non-resonant rectangular cavity adapted to besymmetrically fed the electromagnetic wave by the open ended rectangularwaveguide, the non-resonant rectangular cavity adapted to be filled witha substance of interest; a network analyzer; and a system to determinethe complex dielectric constant of the substance of interest, the systemincluding a processor and a memory, the processor executing instructionsof a program stored in the memory to implement the determination of thecomplex dielectric constant, the program directing the processor toperform the following: (a) calculate a first variable LL as${{LL} = {{\sum\limits_{m = 1}^{\;}{\sum\limits_{n = 1}^{\;}{{B_{mn}\left( \frac{n\;\pi}{b} \right)}{I_{yg}\left( {m,n} \right)}}}} - {\frac{1}{{j\omega\mu}_{o}}{\sum\limits_{m = 1}^{\;}{\sum\limits_{n = 1}^{\;}{A_{mn}{\gamma_{mn}\left( \frac{m\;\pi}{a} \right)}{I_{yg}\left( {m,n} \right)}}}}} - {\frac{1}{{j\omega\mu}_{o}}{\sum\limits_{m = 2}^{\;}{A_{m\; 0}{\gamma_{mo}\left( \frac{m\;\pi}{a} \right)}{I_{yg}\left( {m,o} \right)}}}}}},$where in and n are summation indices ranging between 1 and N, where N isan integer selected for stabilization of the summations,${A_{mn} = {\frac{{- m}\;{\pi/a}}{k_{mn}^{2}}\left( \frac{4}{a\; b} \right){I_{yg}\left( {m,n} \right)}}},$where$k_{mn} = \sqrt{\left( \frac{m\;\pi}{a} \right)^{2} + \left( \frac{n\;\pi}{b} \right)^{2}}$and${{I_{yg}\left( {m,n} \right)} = {\int_{x = 0}^{a}{\int_{y = 0}^{b}{{f_{y}\left( {x,y} \right)}{\sin\left( \frac{m\;\pi\; x}{a} \right)}{\cos\left( \frac{n\;\pi\; y}{b}\  \right)}{\mathbb{d}x}\ {\mathbb{d}y}}}}},$where${{f_{y}\left( {x,y} \right)} = \frac{\sin\left( {\frac{\pi}{a}x} \right)}{\sqrt{1 - \left( \frac{y - \frac{b}{2}}{\frac{b}{2}} \right)^{2}}}},$x and y being Cartesian coordinates corresponding to width and height,respectively,${B_{mn} = {{- \left( \frac{\frac{n\;\pi}{b}}{\frac{m\;\pi}{a}} \right)}\left( \frac{{j\omega\varepsilon}_{o}}{\gamma_{mn}} \right)A_{mn}}},$where j is the imaginary number, ω is an angular frequency of theelectromagnetic wave, ∈_(o) is the constant permittivity of free space,and ${\gamma_{mn} = \sqrt{k_{mn}^{2} - k_{o}^{2}}},$ where${k_{0} = {\omega\sqrt{\mu_{0}\varepsilon_{o}}}},$ where μ₀ is theconstant magnetic permeability of free space, wherein$A_{m\; 0} = {\frac{- 2}{m\;{\pi/\alpha}}\left( \frac{1}{a\; b} \right){I_{y\; g}\left( {m,0} \right)}}$for m≠1; (b) establish a first set of estimated values for a real partof a dielectric constant associated with the substance of interest,∈_(i), where i ranges between 1 and a pre-selected integer S; (c)calculate a second variable M_(i) for i ranging between 1 and S as${M_{i} = {{\sum\limits_{p = 1}{\sum\limits_{q = 1}^{\;}{{C_{pq}\left( \frac{q\;\pi}{h} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}^{\;}{\sum\limits_{q = 1}^{\;}{D_{pq}{\alpha_{pq}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}^{\;}{D_{p\; 0}{\alpha_{po}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{p\; 0}d} \right)}{I_{yc}\left( {p,o} \right)}}}}}},$where p and g are summation indices ranging between 1 and N,${D_{pq} = {\frac{{- p}\;{\pi/g}}{k_{pq}^{2}{\sin\left( {\alpha_{pq}d} \right)}}\left( \frac{4}{g\; h} \right){I_{yc}\left( {p,q} \right)}}},$where$k_{pq} = \sqrt{\left( \frac{p\;\pi}{g} \right)^{2} + \left( \frac{q\;\pi}{h} \right)^{2}}$and $\alpha_{pq} = \sqrt{{k_{i}^{2} - k_{pq}^{2}},}$ where$k_{i} = {\omega\sqrt{{\mu_{0}\varepsilon_{o}ɛ_{i}},}}$ and${C_{pq} = {{- \left( \frac{\frac{q\;\pi}{h}}{\frac{p\;\pi}{g}} \right)}\left( \frac{\left( {{j\;{\omega ɛ}_{o}ɛ_{i}} + \delta} \right)}{\alpha_{pq}} \right)D_{pq}}},$where${I_{yc}\left( {p,q} \right)} = {\int_{x = 0}^{a}{\int_{y = 0}^{b}{{f_{y}\left( {x,y} \right)}{\sin\left( \frac{p\;\pi\; x}{g} \right)}{\cos\left( \frac{q\;\pi\; y}{h} \right)}{\mathbb{d}x}{\mathbb{d}y}}}}$and${D_{p\; 0} = {\frac{- 1}{\frac{p\;\pi}{g}{\sin\left( {\alpha_{p\; 0}d} \right)}}\left( \frac{2}{g\; h} \right){I_{yc}\left( {p,0} \right)}}};$(d) establish a first set of values of ∈_(i) corresponding to thecalculated second variable M_(i) which satisfy the condition${\frac{1 - R_{1}}{1 + R_{1}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}},$where R₁ is a first reflection coefficient of a transmittedelectromagnetic wave measured at the interface between the open endedrectangular waveguide and the non-resonant rectangular cavity at a firstcross-sectional length d by the network analyzer; (e) establish a secondset of values of ∈_(i) corresponding to the calculated second variableM_(i) which satisfy the condition${\frac{1 - R_{2}}{1 + R_{2}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}},$where R₂ is a second reflection coefficient measured at the interfacebetween the open ended rectangular waveguide and the non-resonantrectangular cavity at a second cross-sectional length d different fromthe first cross-sectional length d by the network analyzer; (f) select avalue of ∈_(i) which is a member of both the first set and the secondset of values corresponding to the calculated second variable ∈_(i) andset this selected value equal to a variable ∈_(d); (g) establish asecond set of estimated values for the real part of the dielectricconstant associated with the substance of interest, ∈_(ri), where iranges between 1 and S, such that ∈_(ri)=∈_(i)−j∈′, where ∈′ is animaginary part of the dielectric constant associated with the substanceof interest, where ∈_(r1)=∈_(d)−x₁ and ∈_(rS)=∈_(d)+x₁ and ∈′=x₂, wherex₁ is approximately equal to 0.5 and x₂ is a pre-selected value based onexpected conductivity of the substance of interest; (h) re-calculate thesecond variable M_(i) for i ranging between 1 and S as${M_{i} = {{\sum\limits_{p = 1}{\sum\limits_{q = 1}^{\;}{{C_{pq}\left( \frac{q\;\pi}{h} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}^{\;}{\sum\limits_{q = 1}^{\;}{D_{pq}{\alpha_{pq}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{pq}d} \right)}{I_{yc}\left( {p,q} \right)}}}}} + {\frac{1}{j\;{\omega\mu}_{o}}{\sum\limits_{p = 1}^{\;}{D_{p\; 0}{\alpha_{po}\left( \frac{p\;\pi}{g} \right)}{\cos\left( {\alpha_{p\; 0}d} \right)}{I_{yc}\left( {p,o} \right)}}}}}},$where$k_{i} = {\omega\sqrt{\mu_{0}{\varepsilon_{o}\left( {ɛ_{ri} - ɛ^{\prime}} \right)}}}$and${C_{pq} = {{- \left( \frac{\frac{q\;\pi}{h}}{\frac{p\;\pi}{g}} \right)}\left( \frac{\left( {{j\;{\omega ɛ}_{o}ɛ_{ri}} + \delta} \right)}{\alpha_{pq}} \right)D_{pq}}};$(i) establish a first set of values of ∈_(ri) corresponding to there-calculated second variable M_(i) which satisfy the condition${\frac{1 - R_{1}}{1 + R_{1}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}},$(j) establish a second set of values of ∈_(ri) corresponding to there-calculated second variable M_(i) which satisfy the condition${\frac{1 - R_{2}}{1 + R_{2}} = \frac{\left( {{LL} - M_{i}} \right)({ab})j\;{\omega\mu}_{o}}{2\;{I_{yg}^{2}\left( {1,0} \right)}\gamma_{10}}},$(k) select a value of ∈_(ri) which is a member of both the first set andthe second set of values corresponding to the re-calculated secondvariable M_(i) and setting the selected value equal to a variable∈_(rd); and (l) re-calculate the imaginary part of the dielectricconstant associated with the substance of interest, ∈′, as∈′=j(∈_(rd)−∈_(d)), wherein the dielectric constant associated with thesubstance of interest is determined as ∈_(rd)+j(∈_(rd)−∈_(d)).